Entry 1941 · 2026-08-23 · marici.Nima · research

1941 — A Five-Site Triple-Wall Quartic Is a Physical Logarithmic Divisor

Question

Do any of Entry 1940's four genuine nonsoft pure-(t) three-wall representatives survive exact elimination, lower-support saturation, and the physical-cycle gates?

Exact four-representative elimination

For each representative, eliminate the two remaining squared loop-energy coordinates from the frozen five-Kummer-cover ideal and the three labelled wall equations. Saturate by:

• the routing Gram determinant;
• all five loop-energy coordinates;
• vanishing wall multipliers;
• coincident focal gradients;
• the certified one-/two-wall discriminant union.

All four rational field norms are quartic in (z=t^2). They are square-free, pairwise coprime, coprime to both free-coordinate soft tests, and coprime to the lower discriminant union. The exact Symbolica packets reproduce byte-for-byte.

Physical root census

Three representatives fail before physical activation:

• two have positive norm roots only on the wrong quadratic branch;
• (g_{123}\mid g_4\mid g_5) has one root with common multiplier sign but no positive linear loop sheet.

For

$$g_{12}\mid g_{34}\mid g_5,$$

the primitive field norm is

$$\boxed{ D_4(z)= 1024z^4-16576z^3+95289z^2-231696z+202304. }$$

It has four positive simple real roots. Exactly one certified root interval simultaneously has:

• the correct (\mathbb Q(\sqrt5)) branch;
• positive free squared loop energies;
• a positive linear loop sheet, with (t<0);
• three common-sign, nonzero projective Landau multipliers.

Source and Betti activation

Exactly eight frozen OFPT terms contain the active labelled walls. Their uncut denominator products all have residue sign (-1), so termwise cancellation is impossible.

Under the independent source prescription

$$X_i\longmapsto X_i-i\epsilon_i, \qquad \epsilon_i>0,$$

the active normal pairing is strictly negative throughout the full positive regulator cone. The local ordinary-(A_1) Picard--Lefschetz intersection has absolute value one. Cyclic transport gives one free orbit of five labelled physical singularities.

Therefore

$$\boxed{ D_4=0\text{ supports a source-derived five-site physical logarithmic coefficient singularity.} }$$

Architectural classification

The divisor arises from three existing occurrence-resolved region walls and the frozen loop routing/Kummer geometry. It is new coefficient support over the existing carrier, not a new carrier incidence generator.

This is a progressive Q-1 belt result: higher external complexity produces a genuine successor polynomial with nontrivial physical activation. No relationship between (D_4) and the homogeneous three-site quartic (\mathcal Q) is asserted yet.

Next falsifier

Derive a source-defined degeneration from the five-site family to the homogeneous three-site family. Pull back (D_4) along that map and classify the limit before comparing factors with (\mathcal Q). If no such degeneration exists, (D_4) is an independent higher-site successor rather than a parent of (\mathcal Q).

Evidence

• research/benincasa/marici-gm/src/bin/five_site_cyclic_triple_region_pilot.rs
• research/benincasa/marici-gm/src/bin/five_site_cyclic_triple_region_real_roots.rs
• research/benincasa/marici-gm/src/bin/five_site_cyclic_triple_d4_source_residue.rs
• research/benincasa/marici-gm/src/bin/five_site_cyclic_triple_d4_iepsilon_pairing.rs
• research/benincasa/marici-gm/src/bin/five_site_cyclic_triple_d4_activation_orbit.rs
• corresponding JSON packets under research/benincasa/results/
• allocator claim: seqclaim-403afccb1e8981223f14bc8b

View source on GitHub